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Properties

Label 2-152880-1.1-c1-0-99
Degree $2$
Conductor $152880$
Sign $-1$
Analytic cond. $1220.75$
Root an. cond. $34.9392$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s − 5-s + 9-s − 4·11-s − 13-s + 15-s + 6·17-s + 4·19-s − 8·23-s + 25-s − 27-s + 6·29-s − 8·31-s + 4·33-s − 10·37-s + 39-s + 6·41-s − 4·43-s − 45-s − 6·51-s − 10·53-s + 4·55-s − 4·57-s + 4·59-s + 2·61-s + 65-s + 12·67-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.447·5-s + 1/3·9-s − 1.20·11-s − 0.277·13-s + 0.258·15-s + 1.45·17-s + 0.917·19-s − 1.66·23-s + 1/5·25-s − 0.192·27-s + 1.11·29-s − 1.43·31-s + 0.696·33-s − 1.64·37-s + 0.160·39-s + 0.937·41-s − 0.609·43-s − 0.149·45-s − 0.840·51-s − 1.37·53-s + 0.539·55-s − 0.529·57-s + 0.520·59-s + 0.256·61-s + 0.124·65-s + 1.46·67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 152880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 152880 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(152880\)    =    \(2^{4} \cdot 3 \cdot 5 \cdot 7^{2} \cdot 13\)
Sign: $-1$
Analytic conductor: \(1220.75\)
Root analytic conductor: \(34.9392\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 152880,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + T \)
5 \( 1 + T \)
7 \( 1 \)
13 \( 1 + T \)
good11 \( 1 + 4 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + 8 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 + 10 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 10 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 + 16 T + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 - 16 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 + 10 T + p T^{2} \)
97 \( 1 - 6 T + p T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−13.59248796951930, −12.89922808477728, −12.48209089079895, −12.20461165026730, −11.68728959369622, −11.27877973069101, −10.54964942011916, −10.34234898024568, −9.813012607396027, −9.441220832738472, −8.579167593548984, −8.116312816830010, −7.725041366554574, −7.286346899645427, −6.806478575047569, −5.992388513569225, −5.604453608188408, −5.196542499688770, −4.695556409002214, −3.988717559133464, −3.383264016511480, −2.985719917597865, −2.124191168659763, −1.518836180434413, −0.6618939752018018, 0, 0.6618939752018018, 1.518836180434413, 2.124191168659763, 2.985719917597865, 3.383264016511480, 3.988717559133464, 4.695556409002214, 5.196542499688770, 5.604453608188408, 5.992388513569225, 6.806478575047569, 7.286346899645427, 7.725041366554574, 8.116312816830010, 8.579167593548984, 9.441220832738472, 9.813012607396027, 10.34234898024568, 10.54964942011916, 11.27877973069101, 11.68728959369622, 12.20461165026730, 12.48209089079895, 12.89922808477728, 13.59248796951930

Graph of the $Z$-function along the critical line